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Differential Geometry: Showing a curve is a sphere curve

  1. Jan 24, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that a(x) =( -cos(2x)), -2*cos(x), sin (2x)) is a sphere curve by showing that (-1,0,0) belongs to each normal plane.


    2. Relevant equations
    Not quite sure (part of my question)
    T= a'(x)
    N= T'/norm(T')
    B= T x N (T cross N)
    3. The attempt at a solution

    Ok I found the values of the Tangent field, the Binormal Field, and the Normal field vectors to be certain values and have attempted to demonstrate, by finding a value of x for each of the fields, that the point exists for all three of the planes. However, I don't think this is what should have been done. In fact I am frankly confused as to how exactly to go about showing the asked statement is true. So I guess what I am asking for is did I do this right (see work below), or am I in the wrong frame of mind?

    T= (2 sin(2x), 2 sin(x), 2 cos(2x))
    N=( (4 cos(2x), 2 cos(x), -4sin(2x))/(sqrt(cos^2(x)+4)

    (B is kind of nasty to write up without latex...so I will work on that one, but for the time being if you need to figure it out just go with B = T cross N).

    Thanks for any help.
     
  2. jcsd
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